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Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1944g
if this polytope has a name.
Group : SmallGroup(1944,2344)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 27, 81, 81, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6,2}*648b, {6,6,6}*648b
   9-fold quotients : {6,6,2}*216
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)
(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);;
s1 := ( 2, 3)( 4, 6)( 7, 8)(10,27)(11,26)(12,25)(13,20)(14,19)(15,21)(16,22)
(17,24)(18,23)(28,55)(29,57)(30,56)(31,60)(32,59)(33,58)(34,62)(35,61)(36,63)
(37,81)(38,80)(39,79)(40,74)(41,73)(42,75)(43,76)(44,78)(45,77)(46,68)(47,67)
(48,69)(49,70)(50,72)(51,71)(52,66)(53,65)(54,64);;
s2 := ( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)(10,28)
(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,46)(20,48)(21,47)
(22,52)(23,54)(24,53)(25,49)(26,51)(27,50)(55,64)(56,66)(57,65)(58,70)(59,72)
(60,71)(61,67)(62,69)(63,68)(74,75)(76,79)(77,81)(78,80);;
s3 := (28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)
(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)
(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)
(60,63)(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);
s1 := Sym(81)!( 2, 3)( 4, 6)( 7, 8)(10,27)(11,26)(12,25)(13,20)(14,19)(15,21)
(16,22)(17,24)(18,23)(28,55)(29,57)(30,56)(31,60)(32,59)(33,58)(34,62)(35,61)
(36,63)(37,81)(38,80)(39,79)(40,74)(41,73)(42,75)(43,76)(44,78)(45,77)(46,68)
(47,67)(48,69)(49,70)(50,72)(51,71)(52,66)(53,65)(54,64);
s2 := Sym(81)!( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)
(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,46)(20,48)
(21,47)(22,52)(23,54)(24,53)(25,49)(26,51)(27,50)(55,64)(56,66)(57,65)(58,70)
(59,72)(60,71)(61,67)(62,69)(63,68)(74,75)(76,79)(77,81)(78,80);
s3 := Sym(81)!(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)
(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)
(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);
poly := sub<Sym(81)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0 >; 
 
References : None.
to this polytope